Example 2.2 
Let us introduce a simple chemical reaction sub-network with inflow and outflow based on the well-known kinetic model of glycolysis studied in [1]} and is shown in Fig. REF . We consider a sub-network of glycolysis kinetic model that involves the metabolic pathways of 3PGA, 2PGA and PEP while the influence from the other part of the network is considered as inflow and outflow. Fig. REF  depicts this sub-network where \(x_1,x_2\)  and \(x_3\)  represent the metabolite concentrations of 3PGA, 2PGA and PEP, respectively. The constants \(k_1,k_{-1},k_2,k_{-2}, k_{\text{in}}\)  and \(k_{\text{out}}\)  in Fig. REF  are the rate constants, constant inflow and outflow rate constant, respectively. Following the rate constants used in [1]} (excluding the nonlinear kinetic rate components that appear as the common denominator in each kinetic law in [1]}), we will use \(k_1=\frac{V_{\textup {max}}^\textup {PGM}}{K_m^\textup {PGM}}\) , \(k_{-1}=\frac{V_{\textup {max}}^\textup {PGM}}{K_m^\textup {PGM}\times K_{\textup {eq}}^\textup {PGM}}\) , \(k_2=\frac{V_{\textup {max}}^\textup {ENO}}{K_m^\textup {ENO}}\) , \(k_{-2}=\frac{V_{\textup {max}}^\textup {ENO}}{K_m^\textup {ENO}\times K_{\textup {eq}}^\textup {ENO}}\) , \(k_{\text{out}}=\frac{V_{\textup {max}}^\textup {PYK}}{K_m^\textup {PYK}}\)  and \(k_{\text{in}}=V_{\textup {max}}^\textup {PGK}\)  throughout this paper. Using the numerical values in [1]}, we have \(k_1=7.83\) , \(k_{-1}=41.21\) , \(k_2=33.75\) , \(k_{-2}=5.04\) , \(k_{\text{out}}=7.64\)  and \(k_{\text{in}}=4.8\) . However, as these constants do not satisfy the Wegscheider's condition for detailed balanced CRN (see [5]}), e.g., they do not fulfill \(k_1 k_2=k_{-1} k_{-2}\) . Accordingly, we performed least square estimate to satisfy Wegscheider's condition with the above constants as the priors, which results into the following admissible parameters: \(k_1=7.19\) , \(k_{-1}=41.11\) , \(k_2=32.53\) , \(k_{-2}=5.69\) . Using these values, the kinetics of the glycolysis sub-network can be written as (REF ) with
\(\dot{x}=\underbrace{\left[\begin{matrix} -7.19&41.11&0\\7.19&-73.64&5.69\\0&32.53&-13.33 \end{matrix}\right]}_{A}x+\underbrace{\left[\begin{matrix} 4.8\\0\\0 \end{matrix}\right]}_{B}u.\) 
